The model presented here can be used
to test ideas about pedaling a bike. It maintains geometrical relationships,
gives plots of forces acting at the pedals, gives plots of forces powering
the legs, and animates a model of your pedaling. However,
the best virtual world won't turn your pedals.
Referring to Figure 1, the thigh, shin, and, crank
length together with the position of the point of rotation of the thigh,
point d in figure 1, define the geometry of the model. (Point d is not the
height of the saddle, but the point where the thigh rotates, the point adjusted
by the height of the saddle.)
The model divides a full rotation of the pedals into a
number of segments for analysis. The shaded area in Figure 1 represents
the work performed by one leg moving through one of these segments. Summing
over all segments (the area under the curve) gives the Work of Revolution
performed by one leg in one revolution of the pedals. Work of Revolution
at a given cadence is related to output power. The area under the Work-of-Revolution
curve shows where in the pedal stroke work is done. Figure 1 show a representation
of the output from the Model. It shows the force vectors acting at
the pedal: the tangent force at the crank circle is in green and the radial
force at the crank circle is in red. The green arcs on the thigh and shin
represent the strength and direction of the muscles moving the thighs and
shins (labeled thigh and shin moments) and are shown to scale in model output.
Powering the Pedals, Powering the Model
A torque is a tendency to rotate. Muscles rotate
the thighs about the hip and rotate the shin about the knee. The strength
and speed of the motions of the thighs and shins power the pedals.
Thighs and shins can move in two directions. Muscles that bend the
thigh and shin relative to the torso are called thigh and shin flexors.
Muscles that move the thighs and shins in the opposite direction are called
thigh and shin extensors. The torque produced by the thigh or shin can be
different at each point in the range of motion and depends on the speed
of motion. The torque at each point is described by a function here
called a strength function. The strength and direction of the thigh
and shin extensors are represented by green arcs on the thigh and shin in
Figure 1. (Some readers may think of these strength functions
as strength curves or moment functions or couples.)
Figure 2, A sample plot of
Thigh Extensor and Flexor Strength Functions. These functions are active,
in this example, over a range of motion of about 20 to 60 degrees and show
Thigh Extensor with a maximum value near 240 N m and a Thigh Flexor with
a maximum value of 20 N m but in a direction of motion (negative direction)
opposite to the Extensor. One can see from the example plot in Figure 2
that "pushing," (thigh extensor) has a greater torque than "pulling,"
(thigh flexor motion), an order of magnitude difference.
Strength functions power the model. The geometry defines
the range of motions through which these functions act. The functions describe
the torque (force x distance) available from the muscles at each point in
a range of motion. The shape of the strength function describes how much
and how quick a rider can put power into the pedals.
In concept, a rider can measure torques using testing machines
available at sports medicine clinics. Each of the four motions should be
isolated for each leg, and measurements should be taken at speeds of motion
corresponding to specific cadences under study. Measurements should be taken
for ranges of motion specific to the geometry. The model can furnish these
ranges of motion for use in collection of torque data. The dots in the example
plot of Thigh Moment Functions are experimental data values. The curves
are strength functions fitted by the model to these data.
Figure 3. The shape of the
strength function and its maximum value depend on cadence. At fast cadence,
one could expect the maximum to be less and to have a shape different from
functions at slower cadences. What are the optimum cadence and strength
functions to maximize power? Can riders be trained to improve their strength
functions? During a sprint one could theorize that the strength function
for the thigh extensor becomes very pointed and narrow (an example shown
in Figure 3). What is the power from such a curve? Different geometries
can change the range of motion of the strength functions and their shapes.
"So, if one works one's quads very hard in the gym
how much more power can be expected next season?" Said another way,
"If one improves the quads (shin extensor strength function) by 50%,
how is power affected?"
Parameters Calculated by Pedal Model
The Pedal Model calculates the torque at the bottom bracket.
This torque integrated (in the case of the model, summed over a the number
of segments of rotation) over one complete rotation yields the Work of Revolution.
For a given cadence, power in watts is calculated from the Work of Revolution.
Plots of radial and tangential forces at the pedal, torques about the bottom
bracket, and Work of Revolution are provided. Thigh and shin strength functions
are plotted. An animation of pedaling showing forces and strength functions
Uses of Model
The model can be used to test the effects of changes in
position, crank length, and shin length (heel up or down can effectively
change "shin length"). It can be used to test the effects of different
strength functions (better pedaling techniques, stronger muscle development).
How do longer cranks change power output? How is power affected by changing
seat position? Do people with shorter shins have an advantage? Different
shapes of the strength functions produce different power outputs. A wider
range of motion could give more power. Different regions within the full
range of motion could yield different power outputs.
The "shin length" is taken to be the distance
from the point of rotation of the knee to the center of the pedal axle.
"Shin length" can thus be changed by pointing the heel up or down.
Is there an advantage to keeping one's heel
The angle between the pelvis and horizontal (difficult
to measure) combined with the angle between the horizontal and the thigh
(easy to measure) define the range of motion of the thigh. The ranges of
thigh motion given by the Pedal Model are the angle between the thigh and
horizontal. This is not the same as the angle between the thigh and some
point on the pelvis. It's this angle between some point on the pelvis and
the thigh that one needs to measure if one expects to do weight training
or collect thigh strength function data in the same range of motion as one
The point of rotation of the thighs at the hip (see point
d in Figure 1) is not the same as saddle height; although, this point of
rotation can be adjusted by changing the saddle height.
The model presented here neglects the weight of the legs.
The Moment Functions power the model, and this power is used to propel a
rider and bike and to also move the legs. The movement of the legs would
seem to be a significant part of the total effort. Whether Moment Functions
work against torque at the bottom bracket or the weight of the legs, they
are still doing the work required to move the rider down the road. Measuring
the Moment Functions as independent components and using the pedal model
to analyze alternatives should give ways of producing maximum power from
Copyright ©1990-1997 Tom